Question: Let $f(x) = 3x^{2}-4x-9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
Answer: The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $3x^{2}-4x-9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 3, b = -4, c = -9$ $ x = \dfrac{+ 4 \pm \sqrt{(-4)^{2} - 4 \cdot 3 \cdot -9}}{2 \cdot 3}$ $ x = \dfrac{4 \pm \sqrt{124}}{6}$ $ x = \dfrac{4 \pm 2\sqrt{31}}{6}$ $x =\dfrac{2 \pm \sqrt{31}}{3}$